Continue Game Play
In the Classic version, there is no apparent continuation mode. However, when you get a 'Game Over' go back to the screen where you start on level 1 and scroll down. You will see the Bubble Bobble dinosaur go down on a elevator-type platform to stop at the last level you were at. For example, if you died at level 14 then you can continue again at that point after the 'Game Over'.

Power Up! Mode (Classic Version)
At the ''Classic'' Mode title screen enter: Select, R, L, Left, Right, R, Select, Right. This is the same effect as collecting shoes (Move faster, make bubbles faster)

Secrets
Accessing final 10 levels (in New Version)
To access the final ten levels, you have to collect 9 letters. To get these, kill at least 4 enemies in one Super Bubble, and collect the key icon that is dropped. Then, carry on until the boss of that zone (levels ending in 0) and kill it. Instead of dropping a giant gem, it will leave a letter. Pick up nine of these (one per boss) to ultimately finish the game.

Unlock Robolun (New version)
To unlock Robolun, finish all 100 levels in the new version, and select him from the character select menu. He moves faster than Bobblun and Bubblun, but can't jump as high.

Unlockables
Unlock Lovelun (New Version)
Complete the new version with Robolun to unlock the final secret character, Lovelun. She can fire bubbles faster and further than the other three characters.